What is a hypothesis

A falsifiable statement about what we believe will happen based on the theory we are trying to test.

Falsifiability

• We start from the assumption that our theory is wrong

Falsifiable

• We start from the assumption that our theory is wrong
• We assume there is no relationship

Falsifiable

• We start from the assumption that our theory is wrong
• We assume there is no relationship
• Our hypothesis is called the

alternative hypothesis or H1

Falsifiable

• We start from the assumption that our theory is wrong
• Our hypothesis is called the

alternative hypothesis or H1

• because it is the alternative to our assumption of no relationship

Falsifiable

• We start from the assumption that our theory is wrong
• Our hypothesis is called the:

alternative hypothesis or H1

• because it is the alternative to our assumption of no relationship. This is the:

null hypothesis or H0

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis
• They show the likelihood that the alternative hypothesis is due to random chance

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis
• They show the probability that the alternative hypothesis is due to random chance
• When the probability, p, is below our pre-determined threshold, we reject the null hypothesis

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis
• They show the probability that the alternative hypothesis is due to random chance
• When the probability, p, is below our pre-determined threshold, we reject the null hypothesis
• If we reject the null hypothesis, the alternative hypothesis is true, right?

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis
• If we reject the null hypothesis, the alternative hypothesis is true, right?

NO!!!!

Rejecting the null hypothesis

If we reject the null we infer that the alternative hypothesis is approximately true within the probability we have chosen.

How do we get there?

How do we get from a sample with a correlation to talking about testing a hypothesis for a population?

How do we get there?

How do we get from a sample with a correlation to talking about testing a hypothesis for a population?

• Probability distributions
• Tying our data to the probability distributions
• Tying sample statistics to population parameters

Probability distributions

68-95-99.7 Rule

Probability distributions

68-95-99.7 Rule

            + Allows us to estimate probability based on distance from the mean

Probability distributions

68-95-99.7 Rule

            + Allows us to estimate probability based on distance from the mean
            + Applies to normal distribution
            

Probability distributions

The 68-95-99.7 Rule

            + Allows us to estimate probability based on distance from the mean
            + Applies to normal distribution
            + Basis for the actual decision rules
            

The 68-95-99.7 Rule

68-95-99.7 rule

Source:https://towardsdatascience.com/understanding-the-68-95-99-7-rule-for-a-normal-distribution-b7b7cbf760c2

Populations and Samples

            + Population - The entire group we want to draw conclusions about

Populations and Samples

            + Population - The entire group we want to draw conclusions about
            + Sample - The subset of the population that we draw data from
            

Populations and Samples

            + Population - The entire group we want to draw conclusions about
            + Sample - The subset of the population that we draw data from
            + The sample is a random subset of the population

Populations and Samples

            + Population - The entire group we want to draw conclusions about
            + Sample - The subset of the population that we draw data from
            + The sample is a random subset of the population
            + A good sample is representative of the population
            

Populations and Samples

            + Population - The entire group we want to draw conclusions about
            + Sample - The subset of the population that we draw data from
            + The sample is a random subset of the population
            + A good sample is representative of the population

Getting from sample statistic to population estimate

Two tools tie sample statistics to estimates of the true population parameters: standard error and z-score

Getting from sample statistic to population estimate

Two tools tie sample statistics to estimates of the true population parameters: standard error and z-score

            + The standard error is a special case of the standard deviation
            

Getting from sample statistic to population estimate

Two tools tie sample statistics to estimates of the true population parameters: standard error and z-score

            + The standard error is a special case of the standard deviation
            + The z-score is a fairly simple math problem involving subtracting two numbers and dividing by the standard error
            

Getting from sample statistic to population estimate

Two tools tie sample statistics to estimates of the true population parameters: standard error and z-score

            + The standard error is a special case of the standard deviation
            + The z-score is a fairly simple problem involving subtracting two numbers and dividing by the standard error
            + Bonus: the z-score is one of our hypotheses test values for large sample sizes
    

Getting from sample statistic to population estimate

Two tools tie sample statistics to estimates of the true population parameters: standard error and z-score

            + The standard error is a special case of the standard deviation
            + The z-score is a fairly simple problem involving subtracting two numbers and dividing by the standard error
            + Bonus: the z-score is one of our hypotheses test values for large sample sizes
            + Extra bonus: the cutoff point for the z-score in a hypothesis test is really easy to remember

Two important rules

Two rules to tie the sample to probability distributions and population estimates:

            + **The Central Limit Theorem** 
            + **The Law of Large Numbers**
            

Central Limit Theorem

• For a large number of trials, the means of the trials approach a normal distribution regardless of the underlying distribution of the data

Central Limit Theorem

• For a large number of trials, the means of the trials approach a normal distribution regardless of the underlying distribution of the data
• This means that for a sufficient number of trials, we can apply the normal distribution to the sample means.

Central Limit Theorem

• For a large number of trials, the means of the trials approach a normal distribution regardless of the underlying distribution of the data
• This means that for a sufficient number of trials, we can apply the normal distribution to the sample means.
• This allows us to apply the 68-95-99.7 rule!

Central Limit Theorem

• Also tells us that:
• The mean of the sampling distribution will be equal to the mean of the population distribution:
• \(\bar{x} = \mu\)
• The standard deviation of the sampling distribution will be equal to the standard deviation of the population distribution divided by the sample size:
• \(s = \frac{\sigma}{\sqrt{n}}\)

Central Limit Theorem Simulation with Uniform Distribution Data

Central Limit Theorem Simulation with Uniform Distribution Data

Central Limit Theorem Simulation with Uniform Distribution Data

Central Limit Theorem Simulation with Uniform Distribution Data

Law of Large Numbers

• Averages from independent, identically distributed samples converge to the population means that they are estimating.

Law of Large Numbers

• Averages from independent, identically distributed samples converge to the population means that they are estimating.
• In its strongest form, the law states that this “almost surely” happens.

Law of Large Numbers

• Averages from independent, identically distributed samples converge to the population means that they are estimating.
• In its strongest form, the law states that this “almost surely” happens.
• This means that for a sufficiently large sample size, we can assume with a degree of certainty that our sample statistics accurately represent the population parameters!

Law of Large Numbers simulation using coin flips

The mean of a single flipped coin is 0 

The mean of two flips is 0.5 

The mean of ten flips is 0.3 

The mean of twentyfive flips is 0.44 

The mean of twenty five thousand flips is 0.5023333 

What’s this tell us about sample size?

• The CLT begins to apply at a sample size around 30

• The sample size we need is determined by a number of things including the degree of certainty we are looking for

• Want to do some polling? This is where margin of error comes from:

margin of error

Authorship, License, Credits

• Stephen Moore code used to simulate the Law of Large Numbers

• Author: Tom Hanna

• Website: tomhanna.me

• License: This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.</>